**1. Introduction**

The problems associated with the boundary layer mechanism and heat transfer flow through stretched subsurface have been eminently accepted through analysts as long as the presence in structures of enormous industrial and technologically significance. Few of the advanced spreading applications encompass the designing of plastic layers and copper cables, glass-fiber fabricating, food and polymer refining, geothermal power extraction, liquefying-spinning productions, polymer melting, hot roll glass blasting, in the formation of the final product, in the textile industry, and other abundant utilities. Sakiadis [1] performed the developing e ffort in the area of boundary layer flow on a continued stable subsurface flowing with steady velocities. Later, Crane [2] was the earliest who extended the conception for boundary layer flow through stretchable surfaces. He examined a closed mode result for the Newtonian fluid flow past a flat stretched subsurface. Banks [3] investigated the similarity solutions for the boundary layer flow through a stretched wall with non-Newtonian fluid. Gupta and Gupta [4]

broadened the investigated idea by Crane with the heat and mass transfer past a stretchable surface, along with the influence of suction/blowing. Bujurke et al. [5] discussed the heat transfer phenomenon past a boundary layer, along with interval heat generation. Cortell [6] analyzed the viscous fluid flow numerically with heat transfer on a nonlinearly stretchable subsurface. Shahzad et al. [7] developed the exact solutions of the axisymmetric flow with heat transfer for MHD viscous fluid on a nonlinearly radial pervious stretched surface. Hayat et al. [8] explored the MHD axisymmetric flow for third-grade fluid with heat transfer over stretchable sheets. Shateyi and Makinde [9] recorded the heat transfer analysis for a viscous, electrically conducting fluid flow through a radial stretched and convectively heated disk. Khan et al. [10] discussed the mix convection heat transfer to Sisko fluid past radial stretchable surface together with the influence of convection boundary conditions, thermal radiation, and viscous dissipative terms. Since it is known that the standard of the final product relies on the rate of heat transfer as acknowledged, hence the nanofluids have a higher thermal conductivity of the nanoparticles utilized to enhance the rate for heat transfer [11,12]. For this purpose, distinct techniques are adopted to raise the thermal conductivity of the fluids by providing suspension of nano/micro or large-sized particles into liquids. An inventive approach to enhance the heat transfer rate is performed by utilizing nano-scale particles into the governing fluid by Choi et al. [13]. They recorded that by adding a tiny extent (less than 1% by volume) of the nanoparticles to regular heat transfer fluids enhanced the thermal conductivity for the fluids up to almost 4-times and higher. Kuznetsov and Nield [14] discussed the natural convection into a nanofluid through a vertical surface, along with the impact of thermophoresis and the Brownian-motion. Noghrehabadadi et al. [15] explored the heat transfer of nanofluids past a stretched subsurface with supposing of thermal convectively boundary conditions and partial slip. Zaraki et al. [16] analyzed the influence of the various shapes, sizes, and types of nanoparticles, and base-fluid flowing and heat transferring properties for a naturally convective boundary layer.

The investigations for magnetohydrodynamics have significant utilities, and also uses in cooling of nuclear reactors by the induction flow meter and liquid, depending on the capability variation into the fluid in order normal to the flow and the magnetic field. Ferdows et al. [17] explored the problem for magnetized nanofluid mixed convective flow past an exponential stretched plate. Bidin and Nazar [18] discussed the numerical investigation for boundary layer flow through an exponential stretchable surface, along with thermal emission. Khan et al. [19] studied the unsteady boundary layer flow for a nanofluid on a horizontal stretched plate together with the impact of MHD and thermal radiation. Mabood et al. [20] studied the MHD boundary layer nanofluid flowing with the influence of heat transfer and viscous dissipation through a nonlinear stretched surface. Freidoonimehr et al. [21] studied the magnetized stagnation point flow through a stretched/shrinkable surface alongside the impact of chemical reactions and heat absorption/generation. It is conclusive to mention here that Makinde and Animasaun [22] investigated an admirable work related to magnetized nanofluid flow alongside bioconvection with quartic autocatalysis chemical reaction. The results show that for a fixed numeric of a magnetic parameter, the local skin friction further develops at larger thickened parametric value, whereas the rate for local heat transfer turns lesser at a high-temperature parametric value past an uppermost subsurface of a paraboloid of an uprising. The possible developments and/or applications of the presented analysis to the same topic and to other related topics can be seen in [23–38].

The terminology bioconvection was first acknowledged in an article belonging to James Henry Platt to bring about other researchers to this consideration side towards the physics of streaming forms noticed in impenetrable fashions of free-floating microorganisms. In light of Platt [39], the movement of polygonal forms in impenetrable fashions of Tetrahymena (i.e., ciliate and flagellate), such as Benard cells, though not by thermo-convection. Since, it is well-known that the presence of microorganisms (bacteria) are everywhere, and it is illustrious evidence that a large number of bacteria may be accidentally su ffered and sometimes can be shot down when periodically bared to a higher temperature, conflicting that thermophile is an organism that usually can be seen in di fferent heated territories on the earth. The self-impelled motile microorganisms brought enhancement in the density of the base fluid in

a peculiar way to produce a bioconvection kind of stream. Basing on the cause of propulsion, the motile microorganisms perhaps categorized into various types of microorganisms, counting oxytactic or chemotaxis, gyrotactic microorganisms, and negate gravitaxis. Contrasting the motile microorganisms, the nanoparticles are not self-propelled, and their migration is through the Brownian-motion and thermophoresis impact inward nanofluid. Ghorai and Hill [40] farther elucidated that bioconvection is a known terminology to indicate the phenomena for impromptu arrangemen<sup>t</sup> in the suspension of microorganisms, such as algae and bacteria. Bioconvection also can be explained as the macroscopic convective movement of fluid as a result of the density gradient, and is brought about by the jointly floating of motile microorganisms. Alike naturally convective process, bioconvection is induced by versatile stratification density. Kuznetsov and Avramenko [41] interpreted that when bioconvection takes place, it boosts mingling and diminishes the establishing of the particles that are decisive in medicine utilities. Khan and Makinde [42] examined nanofluid bioconvection caused by gyrotactic microorganisms and noticed that the self-propelled motile microorganisms enhance the density of the base-fluid as floating/swimming in a specific manner. Recently, Raees et al. [43] interpreted that bioconvection into nanofluids has enormous contributions in Colibri micro-volumes spectrometer and benefits the stability in nanofluids. Natural convection with double-di ffusive e ffects over a boundary layer nanofluid flow has been examined by Kuznetsov and Nield [44]. Nonetheless, if the stimulators past the subsurface are more imperative and associate to the bulk-fluid, comprising 36 nm nanoparticles and gyrotactic microorganism, alike chemical backlash could be examined by applying the conception of homogeneous–heterogeneous quartic strategy. Sivaraj et al. [45] have discussed the gyrotactic microorganisms on the mechanism of 29 nm copper water nanofluids propagated through a horizontal surface of paraboloid. Amirsom et al. [46] have considered the movement of microorganisms on a magnetized nanofluid in the presence of second order slip conditions via bvp4c computational scheme. Waqas et al. [47,48] used a shooting method to discuss the propagation of nanoparticles and gyrotactic microorganisms through a stretching surface with magnetic and porous e ffects using non-Newtonian fluid models. A few other inquiries on gyrotactic microorganisms can be read here [49–51].

The impulsion of the current investigation is to explore the impact of a non-uniform magnetic field on the conduct of water suspension comprising nanoparticles and motile gyrotactic microorganisms flowing through a stretchable permeable sheet by employing Successive Local Linearization Method (SLLM) with the combination of Chebyshev spectral linearization method [52] not ye<sup>t</sup> available in the existing literature. The governing flow equations and the boundary conditions were brought towards nonlinear ordinary di fferential equations by utilizing the similarity variable transformations, and are than solved numerically by spectral approach.
